Factor the following expression: $-9$ $x^2$ $-19$ $x+$ $24$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-9)}{(24)} &=& -216 \\ {a} + {b} &=& & & {-19} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-216$ and add them together. Remember, since $-216$ is negative, one of the factors must be negative. The factors that add up to ${-19}$ will be your ${a}$ and ${b}$ When ${a}$ is ${8}$ and ${b}$ is ${-27}$ $ \begin{eqnarray} {ab} &=& ({8})({-27}) &=& -216 \\ {a} + {b} &=& {8} + {-27} &=& -19 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-9}x^2 +{8}x {-27}x +{24} $ Group the terms so that there is a common factor in each group: $ ({-9}x^2 +{8}x) + ({-27}x +{24}) $ Factor out the common factors: $ x(-9x + 8) + 3(-9x + 8) $ Notice how $(-9x + 8)$ has become a common factor. Factor this out to find the answer. $(-9x + 8)(x + 3)$